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You need a new lockset, that part cannot be fixed. That is not a deadlatch. Take the old lockset with you when you go to Home Depot, Lowes, or wherever. it should be easily removed with a couple screws, take the latch too, it is usually held in the edge of the door with two screws too.

Thanks for the comments, guys. I eventually found the problem. It was hard getting the rest of the unit off the door, but once I did, I found that a part was broken. Those tabs hold onto sort of a washer with holes in it, and the one on the inside had broken.

So I shuffled out to Walmart (It was after 10:00) and picked up a new unit. The hole these days has to be bigger than this hole was (my house is from the mid-80s), so I went back to Walmart and bought one of those door hole vice grip hold drilling things (about $11.00).

A little back story: This door has never closed exactly right. Probably the door jamb hole and the door hole are just slightly off or whatever. So the wear of that, over time, is probably what caused this to fail. It definitely looks like a cheap knob.

Most knobs are for 2-3/8" and 2-3/4"... It has been that way for a loooong time. What was the diameter of yours?

The reason I even bring this up... a lower quality knob might be for one or the other. Walmart id known for carrying some of these really poor brands. If it's either of the measurements above, do not bore the door. In fact if you can avoid boring it entirely, DON'T. I would rather see someone get a retro set than re-bore the door. This usually leads to misalignment in both the face bore and the side bore.

I came home from work today and found that this cylinder has come loose from the rest of the door knob assembly. There's one of these on the inside and one on the outside (interior kitchen door). The one on the other side of the door does not come out, because the tabs (on the right of this photo) seem to be holding on to something, or are being held onto by something.

I assumed that I needed to "push and twist" or something, but nothing is working. I knew that Google would be able to help me, but I can't seem to find the name of this particular part. I tried lots of words: cylinder, shaft, sleeve, etc.

I would call that the striker assembly, and I would think if you returned it to where you bought it. It would be replaced on the spot, gratis. I only found this out recently when I went to Rona looking for the same replacement part.

Otherwise, I would note the manufacturer then head to my local Re-store and buy a similar model and strip it for the parts you need.

Former door hardware mechanical engineer here. It looks like part of the chassis for some knob or lever that is removable/reversible. That little sheetmetal spring in the middle top is a clip that would allow you to snap on your knob/lever. A lot of times levers have this so that you can flip it upside down depending on the handing of your door. The sheetmetal bent outwards on the right is probably where it was bent out to make it stick into the latch assembly.